incremental_scc.hpp¶
Return a certificate time for every edge in an incremental graph. Divide the prefix-time interval at its midpoint and compute SCCs using the edges already present there. Edges internal to one SCC recurse into the earlier half; all other edges recurse into the later half after contracting those SCCs. Thus every recursion level is linear in its active graph. An edge with returned time t joins its endpoint components after the first t insertions; applying all such joins reconstructs every SCC partition. edge_count+1 denotes an edge that never joins two distinct components.
Verified by incremental_scc.
离线处理按顺序加入有向边后的强连通变化；适合询问两点最早何时进入同一 SCC。
Implementation¶
#ifndef NOYA_INCREMENTAL_SCC_HPP
#define NOYA_INCREMENTAL_SCC_HPP 1
/// @complexity Time: O((V + E) log E).
/// Space: O(V + E log E).
#include "atcoder/scc.hpp"
#include <algorithm>
#include <cassert>
#include <functional>
#include <tuple>
#include <utility>
#include <vector>
namespace noya {
/// @brief Return a certificate time for every edge in an incremental graph.
/// Divide the prefix-time interval at its midpoint and compute SCCs using the
/// edges already present there. Edges internal to one SCC recurse into the
/// earlier half; all other edges recurse into the later half after contracting
/// those SCCs. Thus every recursion level is linear in its active graph. An
/// edge with returned time t joins its endpoint components after the first t
/// insertions; applying all such joins reconstructs every SCC partition.
/// `edge_count+1` denotes an edge that never joins two distinct components.
inline std::vector<int>
incremental_scc_merge_times(int vertex_count,
const std::vector<std::pair<int, int>> &edges) {
assert(vertex_count >= 0);
int edge_count = int(edges.size());
for (auto [from, to] : edges) {
assert(0 <= from && from < vertex_count);
assert(0 <= to && to < vertex_count);
}
std::vector<int> merge_time(edge_count, edge_count + 1);
std::vector<std::tuple<int, int, int>> active;
active.reserve(edge_count);
for (int id = 0; id < edge_count; id++) {
active.emplace_back(id, edges[id].first, edges[id].second);
}
std::vector<int> new_index(vertex_count, -1);
auto divide = [&](auto &self, std::vector<std::tuple<int, int, int>> data,
int left_time, int right_time) -> void {
if (data.empty() || right_time == left_time + 1) {
return;
}
int middle_time = (left_time + right_time) / 2;
int compressed_size = 0;
for (auto [id, from, to] : data) {
if (new_index[from] == -1) {
new_index[from] = compressed_size++;
}
if (new_index[to] == -1) {
new_index[to] = compressed_size++;
}
}
atcoder::scc_graph graph(compressed_size);
for (auto [id, from, to] : data) {
if (id < middle_time) {
graph.add_edge(new_index[from], new_index[to]);
}
}
auto components = graph.scc();
std::vector<int> component_of(compressed_size);
for (int component = 0; component < int(components.size()); component++) {
for (int vertex : components[component]) {
component_of[vertex] = component;
}
}
std::vector<std::tuple<int, int, int>> earlier;
std::vector<std::tuple<int, int, int>> later;
earlier.reserve(data.size());
later.reserve(data.size());
for (auto [id, original_from, original_to] : data) {
int from = new_index[original_from];
int to = new_index[original_to];
if (id < middle_time && component_of[from] == component_of[to]) {
merge_time[id] = std::min(merge_time[id], middle_time);
earlier.emplace_back(id, from, to);
} else {
later.emplace_back(id, component_of[from], component_of[to]);
}
}
for (auto [id, from, to] : data) {
new_index[from] = -1;
new_index[to] = -1;
}
self(self, std::move(earlier), left_time, middle_time);
self(self, std::move(later), middle_time, right_time);
};
divide(divide, std::move(active), 0, edge_count + 1);
return merge_time;
}
/// @brief After every insertion, return the sum of weight products over pairs
/// of vertices in the same strongly connected component.
template <class T>
std::vector<T> incremental_scc_pair_product_sums(
const std::vector<T> &vertex_weights,
const std::vector<std::pair<int, int>> &edges) {
int vertex_count = int(vertex_weights.size());
int edge_count = int(edges.size());
auto merge_time = incremental_scc_merge_times(vertex_count, edges);
std::vector<std::vector<int>> joins(edge_count + 1);
for (int id = 0; id < edge_count; id++) {
if (merge_time[id] <= edge_count) {
joins[merge_time[id]].push_back(id);
}
}
std::vector<int> parent(vertex_count, -1);
std::vector<T> component_sum = vertex_weights;
auto leader = [&](auto &self, int vertex) -> int {
if (parent[vertex] < 0) {
return vertex;
}
return parent[vertex] = self(self, parent[vertex]);
};
T total{};
std::vector<T> result(edge_count);
for (int time = 1; time <= edge_count; time++) {
for (int id : joins[time]) {
int first = leader(leader, edges[id].first);
int second = leader(leader, edges[id].second);
if (first == second) {
continue;
}
total += component_sum[first] * component_sum[second];
if (parent[first] > parent[second]) {
std::swap(first, second);
}
parent[first] += parent[second];
parent[second] = first;
component_sum[first] += component_sum[second];
}
result[time - 1] = total;
}
return result;
}
} // namespace noya
#endif // NOYA_INCREMENTAL_SCC_HPP
#include <algorithm>
#include <cassert>
#include <functional>
#include <tuple>
#include <utility>
#include <vector>
/// @complexity Time: O((V + E) log E).
/// Space: O(V + E log E).
namespace atcoder {
namespace internal {
template <class E> struct csr {
std::vector<int> start;
std::vector<E> elist;
explicit csr(int n, const std::vector<std::pair<int, E>>& edges)
: start(n + 1), elist(edges.size()) {
for (auto e : edges) {
start[e.first + 1]++;
}
for (int i = 1; i <= n; i++) {
start[i] += start[i - 1];
}
auto counter = start;
for (auto e : edges) {
elist[counter[e.first]++] = e.second;
}
}
};
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
// Reference:
// R. Tarjan,
// Depth-First Search and Linear Graph Algorithms
struct scc_graph {
public:
explicit scc_graph(int n) : _n(n) {}
int num_vertices() { return _n; }
void add_edge(int from, int to) { edges.push_back({from, {to}}); }
// @return pair of (# of scc, scc id)
std::pair<int, std::vector<int>> scc_ids() {
auto g = csr<edge>(_n, edges);
int now_ord = 0, group_num = 0;
std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
visited.reserve(_n);
auto dfs = [&](auto self, int v) -> void {
low[v] = ord[v] = now_ord++;
visited.push_back(v);
for (int i = g.start[v]; i < g.start[v + 1]; i++) {
auto to = g.elist[i].to;
if (ord[to] == -1) {
self(self, to);
low[v] = std::min(low[v], low[to]);
} else {
low[v] = std::min(low[v], ord[to]);
}
}
if (low[v] == ord[v]) {
while (true) {
int u = visited.back();
visited.pop_back();
ord[u] = _n;
ids[u] = group_num;
if (u == v) break;
}
group_num++;
}
};
for (int i = 0; i < _n; i++) {
if (ord[i] == -1) dfs(dfs, i);
}
for (auto& x : ids) {
x = group_num - 1 - x;
}
return {group_num, ids};
}
std::vector<std::vector<int>> scc() {
auto ids = scc_ids();
int group_num = ids.first;
std::vector<int> counts(group_num);
for (auto x : ids.second) counts[x]++;
std::vector<std::vector<int>> groups(ids.first);
for (int i = 0; i < group_num; i++) {
groups[i].reserve(counts[i]);
}
for (int i = 0; i < _n; i++) {
groups[ids.second[i]].push_back(i);
}
return groups;
}
private:
int _n;
struct edge {
int to;
};
std::vector<std::pair<int, edge>> edges;
};
} // namespace internal
} // namespace atcoder
namespace atcoder {
struct scc_graph {
public:
scc_graph() : internal(0) {}
explicit scc_graph(int n) : internal(n) {}
void add_edge(int from, int to) {
int n = internal.num_vertices();
assert(0 <= from && from < n);
assert(0 <= to && to < n);
internal.add_edge(from, to);
}
std::vector<std::vector<int>> scc() { return internal.scc(); }
private:
internal::scc_graph internal;
};
} // namespace atcoder
namespace noya {
/// @brief Return a certificate time for every edge in an incremental graph.
/// Divide the prefix-time interval at its midpoint and compute SCCs using the
/// edges already present there. Edges internal to one SCC recurse into the
/// earlier half; all other edges recurse into the later half after contracting
/// those SCCs. Thus every recursion level is linear in its active graph. An
/// edge with returned time t joins its endpoint components after the first t
/// insertions; applying all such joins reconstructs every SCC partition.
/// `edge_count+1` denotes an edge that never joins two distinct components.
inline std::vector<int>
incremental_scc_merge_times(int vertex_count,
const std::vector<std::pair<int, int>> &edges) {
assert(vertex_count >= 0);
int edge_count = int(edges.size());
for (auto [from, to] : edges) {
assert(0 <= from && from < vertex_count);
assert(0 <= to && to < vertex_count);
}
std::vector<int> merge_time(edge_count, edge_count + 1);
std::vector<std::tuple<int, int, int>> active;
active.reserve(edge_count);
for (int id = 0; id < edge_count; id++) {
active.emplace_back(id, edges[id].first, edges[id].second);
}
std::vector<int> new_index(vertex_count, -1);
auto divide = [&](auto &self, std::vector<std::tuple<int, int, int>> data,
int left_time, int right_time) -> void {
if (data.empty() || right_time == left_time + 1) {
return;
}
int middle_time = (left_time + right_time) / 2;
int compressed_size = 0;
for (auto [id, from, to] : data) {
if (new_index[from] == -1) {
new_index[from] = compressed_size++;
}
if (new_index[to] == -1) {
new_index[to] = compressed_size++;
}
}
atcoder::scc_graph graph(compressed_size);
for (auto [id, from, to] : data) {
if (id < middle_time) {
graph.add_edge(new_index[from], new_index[to]);
}
}
auto components = graph.scc();
std::vector<int> component_of(compressed_size);
for (int component = 0; component < int(components.size()); component++) {
for (int vertex : components[component]) {
component_of[vertex] = component;
}
}
std::vector<std::tuple<int, int, int>> earlier;
std::vector<std::tuple<int, int, int>> later;
earlier.reserve(data.size());
later.reserve(data.size());
for (auto [id, original_from, original_to] : data) {
int from = new_index[original_from];
int to = new_index[original_to];
if (id < middle_time && component_of[from] == component_of[to]) {
merge_time[id] = std::min(merge_time[id], middle_time);
earlier.emplace_back(id, from, to);
} else {
later.emplace_back(id, component_of[from], component_of[to]);
}
}
for (auto [id, from, to] : data) {
new_index[from] = -1;
new_index[to] = -1;
}
self(self, std::move(earlier), left_time, middle_time);
self(self, std::move(later), middle_time, right_time);
};
divide(divide, std::move(active), 0, edge_count + 1);
return merge_time;
}
/// @brief After every insertion, return the sum of weight products over pairs
/// of vertices in the same strongly connected component.
template <class T>
std::vector<T> incremental_scc_pair_product_sums(
const std::vector<T> &vertex_weights,
const std::vector<std::pair<int, int>> &edges) {
int vertex_count = int(vertex_weights.size());
int edge_count = int(edges.size());
auto merge_time = incremental_scc_merge_times(vertex_count, edges);
std::vector<std::vector<int>> joins(edge_count + 1);
for (int id = 0; id < edge_count; id++) {
if (merge_time[id] <= edge_count) {
joins[merge_time[id]].push_back(id);
}
}
std::vector<int> parent(vertex_count, -1);
std::vector<T> component_sum = vertex_weights;
auto leader = [&](auto &self, int vertex) -> int {
if (parent[vertex] < 0) {
return vertex;
}
return parent[vertex] = self(self, parent[vertex]);
};
T total{};
std::vector<T> result(edge_count);
for (int time = 1; time <= edge_count; time++) {
for (int id : joins[time]) {
int first = leader(leader, edges[id].first);
int second = leader(leader, edges[id].second);
if (first == second) {
continue;
}
total += component_sum[first] * component_sum[second];
if (parent[first] > parent[second]) {
std::swap(first, second);
}
parent[first] += parent[second];
parent[second] = first;
component_sum[first] += component_sum[second];
}
result[time - 1] = total;
}
return result;
}
} // namespace noya